Question

Resolve into partial fraction for the following:

`(x^2 - 3)/((x + 2)(x^2 + 1))`

Answer 1

Here the quadratic factor x² + 1 is not factorisable.

Let `(x^2 - 3)/((x + 2)(x^2 + 1)) = "A"/(x + 2) + (("B"x + "C"))/(x^2 + 1)`   ...(1)

Multiply both sides by (x + 2) (x² + 1) we get,

x² – 3 = A(x² + 1) + (Bx + C) (x + 2)

Put x = -2 we get

(-2)² – 3 = [A(-2)² + 1] + 0

4 – 3 = A(4 + 1)

1 = 5A

A = `1/5`

Equating coefficient of x² on both sides of (2) we get

1 = A + B

1 = `1/5` + B

B = `1 - 1/5 = 4/5`

Equating coefficients of x on both sides of (2) we get

0 = 2B + C

0 = `2(4/5)` + C

C = `(-8)/5`

Using A, B, C’s values in (1) we get

`(x^2 - 3)/((x + 2)(x^2 + 1)) = 1/(5(x + 2)) + ((4/5x - 8/5))/(x^2 + 1)`

`= 1/(5(x + 2)) + (4/5(x - 2))/(x^2 + 1)`